In seismic exploration, it is common practice to deploy a large array of geophones on the surface of the earth and to record the vibrations of the earth at each geophone location to obtain a collection of seismic traces. The traces are sampled and recorded for further processing. When the vibrations so recorded are caused by a seismic source activated at a known time and location, the recorded data can be processed by a computer in known ways to produce an image of the subsurface. The image thus produced is commonly interpreted by geophysicists to detect the possible presence of valuable hydrocarbons.
Seismograms are commonly recorded as digital samples representing the amplitude of received seismic reflection signals as a function of time. Since seismograms are usually obtained along a line of exploration on the surface of the earth, the digital samples can be formed into an array (t-x) of seismic traces with each sample in a trace representing the amplitude of the seismic trace as a function of time (t) and horizontal distance (x), When such traces are visually reproduced, by plotting or the like, seismic sections are produced. A seismic section depicts the subsurface layering of a section of the earth. It is the principal tool which the geophysicist studies to determine the nature of the earth's subsurface formation.
While the amplitudes of the received seismic reflection signals contain information about the earth's subsurface, such amplitudes are also influenced by the seismic energy sources and receivers that are used to create and record the seismic signals. Differences in source strength and receiver sensitivity, or calibration, cause random amplitude variations, a noise in the seismic reflection's signals, that must be removed in order to extract useful amplitude information.
The standard method for removing the effects of source and receiver variations is called surface consistent amplitude correction and is discussed in detail in "Surface Consistent Corrections" by M. T. Taner and F. Koehler, Geophysics, v.46, No.1, pages 17-22 (1981). In this method, the seismic data are modeled as a product of four scale factors. These factors are (1) a source performance factor, (2) a receiver performance factor, (3) an offset dependent factor, and (4) a subsurface or common midpoint point factor. In this method, an amplitude factor A is computed for each trace. Assuming that the amplitude factor is a product of the source, receiver, offset, and midpoint factors (denoted S, R, H and K respectively), the natural logarithm of A is the sum of the logarithms of the individual factors: EQU log(A)=log(S)+log(R)+log(H)+log(K).
Denoting the logarithms of these factors by A', S', H' and K' yields a single equation in four unknowns. Each trace provides one equation. The total number of equations is equal to the number of traces in the seismic line. Although the number of traces depends on the recording geometry, generally, the number of traces is equal to the product of the number of sources times the number of receivers. The number of unknown performance factors is equal to the sum of the number of sources, the number of receivers, the number of offsets, and the number of midpoints. This system of equations is solved with an optimization criterion to determine the performance factors that best match the amplitude factors. Amplitude balancing is then accomplished by applying the source and receiver performance factors, determined from the optimization problem, to the individual seismic traces.